More on generalizations of topology of uniform convergence and m-topology on C(X)

Abstract

This paper conglomerates our findings on the space C(X) of all real valued continuous functions, under different generalizations of the topology of uniform convergence and the m-topology. The paper begins with answering all the questions which were left open in our previous paper on the classifications of Z-ideals of C(X) induced by the UI and the mI-topologies on C(X) [5]. Motivated by the definition of the mItopology, another generalization of the topology of uniform convergence, called UI-topology, is introduced here. Among several other results, it is established that for a convex ideal I in C(X), a necessary and sufficient condition for UI-topology to coincide with mI-topology on C(X) is the boundedness of X ? T Z[I] in X. As opposed to the case of the UI-topologies (and mI-topologies) on C(X), it is proved that each UI-topology (respectively, mI-topology) on C(X) is uniquely determined by the ideal I. In the last section, the denseness of the set of units of C(X) in CU(X) (= C(X) with the topology of uniform convergence) is shown to be equivalent to the strong zero dimensionality of the space X. Also, the space X turns out to be a weakly P-space if and only if the set of zero divisors (including 0) in C(X) is closed in CU(X). Computing the closure of CP(X) (={ f ? C(X) : the support of f ? P} where P is an ideal of closed sets in X) in CU(X) and Cm(X) (= C(X) with the m-topology), the results clUCP(X) = CP? (X) (= { f ? C(X) : ?n ? N, {x ? X : | f (x)| ? 1n } ? P}) and clmCP(X) = { f ? C(X) : f.1 ? CP? (X) for each 1 ? C(X)} are achieved.